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Some Applications of Classical PDE in finance
Peter Michael [email protected]
Dipartimento di Matematica and Facolta di Statistica
Universita di Roma I
StockAug2005 – p.1/40
Main aims of this contribution
Joint work with Peter Carr (Bloomberg and Courant) andTai-Ho Wang, National Chung Cheng U., Chia Yi, Taiwan)using Hadamard’s method and Lie’s symmetry groupsin PDE in mathematical finance. To briefly discuss threerecent contributions
Work joint with Peter Carr on multi-asset stochastic localvariance
Work joint with Wang on special solvable quadratic andinverse quadratic interest rate models
Joint work with both Carr and Wang on generating andcharacterizing solvable one-dimensional diffusions
StockAug2005 – p.2/40
Topic 1 Multi-asset variance contracts, Multi-asset
Generalization of multi-variance contracts to many assets.
What is a variance contract?
Such a contract specifies a fixed strike K > 0 and a fixed maturity date T . Just after T thecontract pays
δ(ST − K) d < ST >T ,
where
d < ST >T = local quadratic variation
and where δ(ST − K) is Dirac’s Delta function. For example if dSt = atdWt ,< dST >T = a2
T dt.
Question What is the natural generalization of this contract in the context of a contract on abasket of assets?
Answer: The answer will be related to finding a new contract whose payoff is a cut-off of afundamental solution for an appropriate elliptic equation in n variables.
StockAug2005 – p.3/40
Breeden and Litzenberger
C(St, t, K, T ) = e−r(T−t)E[(ST − K)+]
where : E[(ST − K)+] =
∫
<+
(ST − K)+p(St, t, s, T )ds,
and p(st, t, sT , T ) is the transition probability density for passage from St at time t to ST attime T .
So
CKK(K, T ) = e−r(T−t)E[(ST − K)+)KK ] = e−r(T−t)E[δK(S)]
= e−r(T−t)p(St, t, K, T )
ie. Breeden Litzenberger allows us to recover the transition probability from knowledge of
the quoted market prices of call options.StockAug2005 – p.4/40
Breeden-Litzenberger inn-D
Is there a generalization of Breeden and Litzenberger to n
assets?
• Answer: A positive answer (kind of) appeared in the caseof two assets in the book by Lipton (World Scientific, 2002).The answer involves the Radon transform.• The generalization to the n-asset case is straightforwardand also can be formulated in terms of the Radon transform.• However the answer may be of limited practical interest asit assumes liquid market for index options with a continuumof weights.• We will return to it later, time permitting.
StockAug2005 – p.5/40
Review of Dupire’s construction of a variance contract
For simplicity assume zero rates and no dividends.Under the risk neutral measure we have
dSt = atdWt
This is a stochastic volatility model in that at is allowed todepend on sources of uncertainty above and beyond theuncertainty in the spot level St.
For the quadratic variation we have
d < S >t= a2
t dt
StockAug2005 – p.6/40
Review, p2.
Dupire showed that the local variance
E0
[a2
T |ST ∈ dK]
can be determined from the initial market value of straddlesstruck around K and maturing at T .Let V0(K,T ) be the values of such straddles at time 0. Then
E0
[a2
T |ST ∈ dK]
= 2∂
∂TV0(K,T )
∂2
∂K2 V0(K,T )
Ratio of a calendar spread of straddles to a butterfly spreadof straddles. StockAug2005 – p.7/40
Review p. 3
• To establish his formula, Dupire uses the Tanaka-Meyer theorem to get
|ST − K| = |S0 − K| +∫ T
0sgn(St − K)dSt +
∫ T
0δK(St)a
2t dt
• Next, take risk-neutral expectations
V0(K, T ) = |S0 − K| +∫ T
0E0
[a2
T δ(ST − K)]dt
E0
[a2
T |ST = K]
=E0
[a2
T δ(ST − K)]
E0 [δ(ST − K)]
=︸ ︷︷ ︸
using ∂|S0−K|∂K
=2δ(S0−K)
12E0
[a2
T δ(ST − K)]
∂2
∂K2 V (K, T )
StockAug2005 – p.8/40
Review p. 4
(In summary)
∂V0(K, T )
∂T=
1
2E0
[a2
T |ST ∈ dK] ∂2
∂K2V (K, T )
So
E0
[a2
T |ST ∈ dK]dt
= 2
∂V0(K,T )∂T
∂2
∂K2 V (K, T )
Therefore the initial expectation of the terminal local variance given that ST = K is twice the ratioof a calendar spread to a butterfly spread.
QED
StockAug2005 – p.9/40
Synopsis
It is seen that the key element in Dupire’s proof is that
One dimensional Laplacian F = 2δK(S)
where
F = |ST − K|
Therefore the key was that function |ST − K| is afundamental solution of the Laplacian in 1 − D.
StockAug2005 – p.10/40
Generalization to two assets: Normal Dynamics
First case is where the dynamics is normal as opposed to lognormal
dS1t = atσ1dW1t
dS2t = a2tσ2dW2t t ∈ [0, T ]
StockAug2005 – p.11/40
Generalization to two assets: Normal Dynamics
First case is where the dynamics is normal as opposed to lognormal
dS1t = atσ1dW1t
dS2t = a2tσ2dW2t t ∈ [0, T ]
Here the volatility of the stock Si has an idiosyncratic component σi and a stochasticcomponent at common to both stocks. W1t and W2t are two correlated Brownian motions.The instantaneous variance of dS1t + dS2t at time t.
a2t (σ2
1 + 2ρσ1σ2 + σ22)
StockAug2005 – p.11/40
Generalization to two assets: Normal Dynamics
First case is where the dynamics is normal as opposed to lognormal
dS1t = atσ1dW1t
dS2t = a2tσ2dW2t t ∈ [0, T ]
Here the volatility of the stock Si has an idiosyncratic component σi and a stochasticcomponent at common to both stocks. W1t and W2t are two correlated Brownian motions.The instantaneous variance of dS1t + dS2t at time t.
a2t (σ2
1 + 2ρσ1σ2 + σ22)
The idiosyncratic component of this instantaneous variance is
VI = σ21 + 2ρσ1σ2 + σ2
2
StockAug2005 – p.11/40
Normal case ct’d
Our objective is to synthesize the bivariate local variance a2t VIδ(S1t − K1, S2t − K2)dt.
Note that this payoff is zero unless (S1T , S2T ∈ (dK1, dK2) and in this event a2T VIdt is the
increment in the quadratic variation of S1 + S2.
StockAug2005 – p.12/40
Normal case ct’d
Our objective is to synthesize the bivariate local variance a2t VIδ(S1t − K1, S2t − K2)dt.
Note that this payoff is zero unless (S1T , S2T ∈ (dK1, dK2) and in this event a2T VIdt is the
increment in the quadratic variation of S1 + S2.
Natural candidate to take the place of |S − K| in the 2D case is (up to a constant and anorthogonal transformation) the fundamental solution of the Laplacian in 2 − D.
StockAug2005 – p.12/40
Normal case ct’d
Our objective is to synthesize the bivariate local variance a2t VIδ(S1t − K1, S2t − K2)dt.
Note that this payoff is zero unless (S1T , S2T ∈ (dK1, dK2) and in this event a2T VIdt is the
increment in the quadratic variation of S1 + S2.
Natural candidate to take the place of |S − K| in the 2D case is (up to a constant and anorthogonal transformation) the fundamental solution of the Laplacian in 2 − D.
Ie. g(S1, S2) satisfies
σ21
2
∂2
∂S21
g(S1, S2) + 2ρσ1σ2∂2
∂S1dS2g(S1, S2) +
σ22
2
∂2
∂S22
g(S1, S2) = −VIδ(S1t − K1, S2t − K2)
StockAug2005 – p.12/40
Normal case ct’d
Our objective is to synthesize the bivariate local variance a2t VIδ(S1t − K1, S2t − K2)dt.
Note that this payoff is zero unless (S1T , S2T ∈ (dK1, dK2) and in this event a2T VIdt is the
increment in the quadratic variation of S1 + S2.
Natural candidate to take the place of |S − K| in the 2D case is (up to a constant and anorthogonal transformation) the fundamental solution of the Laplacian in 2 − D.
Ie. g(S1, S2) satisfies
σ21
2
∂2
∂S21
g(S1, S2) + 2ρσ1σ2∂2
∂S1dS2g(S1, S2) +
σ22
2
∂2
∂S22
g(S1, S2) = −VIδ(S1t − K1, S2t − K2)
Solution known to be:
StockAug2005 – p.12/40
Normal case ct’d
Our objective is to synthesize the bivariate local variance a2t VIδ(S1t − K1, S2t − K2)dt.
Note that this payoff is zero unless (S1T , S2T ∈ (dK1, dK2) and in this event a2T VIdt is the
increment in the quadratic variation of S1 + S2.
Natural candidate to take the place of |S − K| in the 2D case is (up to a constant and anorthogonal transformation) the fundamental solution of the Laplacian in 2 − D.
Ie. g(S1, S2) satisfies
σ21
2
∂2
∂S21
g(S1, S2) + 2ρσ1σ2∂2
∂S1dS2g(S1, S2) +
σ22
2
∂2
∂S22
g(S1, S2) = −VIδ(S1t − K1, S2t − K2)
Solution known to be:
g(S1, S2) =1
π√
(1 − ρ2)σ1σ2
ln r(S1, S2)
r(S1, S2) ≡
√√√√
1
1 − ρ2
[(S1 − K1
σ1
)2
− 2ρS1 − K1
σ1
S2 − K2
σ2+
(S2 − K2
σ2
)2]
StockAug2005 – p.12/40
Normal ct’d
We wish to apply Ito’s formula to g just as we did in the 1 − D case for |S − K|. However this isnot possible due to the singularity of g. For now, we illustrate the main idea by proceeding at aformal level. Apply Itô’s formula to the process Gt defined by Gt = g(S1t, S2t) implies:
g(S1T , S2T ) = g(S10, S20) +
∫ T
0
∂
∂S1g(S1t, S2t)
︸ ︷︷ ︸
∼S1,t
r2t
dS1t +
∫ T
0
∂
∂S2g(S1t, S2t)
︸ ︷︷ ︸
∼S2,t
r2t
dS2t
+
∫ T
0a2
t
σ21
2
∂2
∂S21
g(S1t, S2t) + ρσ1σ2∂2
∂S1∂S2g(S1t, S2t) +
σ22
2
∂2
∂S22
g(S1t, S2t)
︸ ︷︷ ︸
−VIδ(S1t−K1,S2t−K2)
dt.(6)
Ito’s formula can be used on a truncated version of g we call gε.
StockAug2005 – p.13/40
Normal ct’d
We wish to apply Ito’s formula to g just as we did in the 1 − D case for |S − K|. However this isnot possible due to the singularity of g. For now, we illustrate the main idea by proceeding at aformal level. Apply Itô’s formula to the process Gt defined by Gt = g(S1t, S2t) implies:
g(S1T , S2T ) = g(S10, S20) +
∫ T
0
∂
∂S1g(S1t, S2t)
︸ ︷︷ ︸
∼S1,t
r2t
dS1t +
∫ T
0
∂
∂S2g(S1t, S2t)
︸ ︷︷ ︸
∼S2,t
r2t
dS2t
+
∫ T
0a2
t
σ21
2
∂2
∂S21
g(S1t, S2t) + ρσ1σ2∂2
∂S1∂S2g(S1t, S2t) +
σ22
2
∂2
∂S22
g(S1t, S2t)
︸ ︷︷ ︸
−VIδ(S1t−K1,S2t−K2)
dt.(7)
Ito’s formula can be used on a truncated version of g we call gε.
Idea: Let A = (ΣtΣ)−1 be the inverse of coefficient matrix. Surround singularity (K1, K2)
by an ellipsoid of radius ε, EK(ε) = {X = (x1, x2) : XtAX = ε}
StockAug2005 – p.13/40
Normal ct’d
We wish to apply Ito’s formula to g just as we did in the 1 − D case for |S − K|. However this isnot possible due to the singularity of g. For now, we illustrate the main idea by proceeding at aformal level. Apply Itô’s formula to the process Gt defined by Gt = g(S1t, S2t) implies:
g(S1T , S2T ) = g(S10, S20) +
∫ T
0
∂
∂S1g(S1t, S2t)
︸ ︷︷ ︸
∼S1,t
r2t
dS1t +
∫ T
0
∂
∂S2g(S1t, S2t)
︸ ︷︷ ︸
∼S2,t
r2t
dS2t
+
∫ T
0a2
t
σ21
2
∂2
∂S21
g(S1t, S2t) + ρσ1σ2∂2
∂S1∂S2g(S1t, S2t) +
σ22
2
∂2
∂S22
g(S1t, S2t)
︸ ︷︷ ︸
−VIδ(S1t−K1,S2t−K2)
dt.(8)
Ito’s formula can be used on a truncated version of g we call gε.
Idea: Let A = (ΣtΣ)−1 be the inverse of coefficient matrix. Surround singularity (K1, K2)
by an ellipsoid of radius ε, EK(ε) = {X = (x1, x2) : XtAX = ε}Glue the solution g to the solution outside the ball to the solution inside the ball using aquadratic function, while matching both first and second derivatives. Glued together functionis called gε
StockAug2005 – p.13/40
Normal ct’d
We wish to apply Ito’s formula to g just as we did in the 1 − D case for |S − K|. However this isnot possible due to the singularity of g. For now, we illustrate the main idea by proceeding at aformal level. Apply Itô’s formula to the process Gt defined by Gt = g(S1t, S2t) implies:
g(S1T , S2T ) = g(S10, S20) +
∫ T
0
∂
∂S1g(S1t, S2t)
︸ ︷︷ ︸
∼S1,t
r2t
dS1t +
∫ T
0
∂
∂S2g(S1t, S2t)
︸ ︷︷ ︸
∼S2,t
r2t
dS2t
+
∫ T
0a2
t
σ21
2
∂2
∂S21
g(S1t, S2t) + ρσ1σ2∂2
∂S1∂S2g(S1t, S2t) +
σ22
2
∂2
∂S22
g(S1t, S2t)
︸ ︷︷ ︸
−VIδ(S1t−K1,S2t−K2)
dt.(9)
Ito’s formula can be used on a truncated version of g we call gε.
Idea: Let A = (ΣtΣ)−1 be the inverse of coefficient matrix. Surround singularity (K1, K2)
by an ellipsoid of radius ε, EK(ε) = {X = (x1, x2) : XtAX = ε}Glue the solution g to the solution outside the ball to the solution inside the ball using aquadratic function, while matching both first and second derivatives. Glued together functionis called gε
StockAug2005 – p.13/40
Normal Ct’d
• Therefore we obtain
VI
4π√4ε2
∫ T
0a2
t1(S1t,S2t)∈EKε
dt
︸ ︷︷ ︸
from approximation to delta function
= gε(S10, S20) − gε(S1T , S2T )
+
∫ T
0
∂
∂S1gε(S1t, S2t)dS1t +
∫ T
0
∂
∂S2gε(S1t, S2t)dS2t.
where VI = σ21 + 2ρσ1σ2 + σ2
2 ,
• Differentiate this result with respect to T
VI
4π√4ε2
a2T 1(S1T ,S2T )∈EK
ε
= − ∂
∂Tgε(S1T , S2T ) +
∂
∂S1gε(S1T , S2T )dS1T +
∂
∂S2gε(S2T , S2T )dS2T
• The left hand side is a payoff that captures an average stochastic volatility in an ε neighborhood
of the strike K = (K1, K2).StockAug2005 – p.14/40
Normal Dynamics
• (from last slide)
VI
4π√4ε2
a2T 1(S1t,S2t)∈EK
ε
= − ∂
∂Tgε(S1T , S2T ) + gε(S1T , S2T )dS1T +
∂
∂S2gε(S2t, S2t)dS2t
• This payoff can be replicated by shorting a calendar spread of contingent ε contingent claims
with maturities T and T + ∆T and by going long ∂gε(S1T ,S2T )∂S1
shares of asset 1 and∂gε(S12,S2T )
∂S2shares of asset 2 during the time interval [T, T + ∆T ]/
• Note the crucial role of the ε approximating payoff. The fundamental solution being unbounded
for n ≥ 2, implies that the untruncated claim would not be a marketable claim.
StockAug2005 – p.15/40
Normal Dynamics: Conclusion
• Once again, as in 1-asset case, take risk-neutral expectations after having differentiated theresult with respect to T and obtain
VI
4π√4ε2
EQ
[
a2T 1(S1t,S2t)∈EK
ε
]
= − ∂
∂TE0 [gε
T ]
• So that, with the same reasoning as before we obtain
E[
a2T | (S1t, S2t) ∈ EK
ε
]
= Q(
(S1t, S2t) ∈ EKε
) (
− ∂
∂TE [−gε
T ]
)
• So if we know the risk-neutral proability that (S1t, S2t) ∈ EKε we have again the result that
the averaged local variance can be captured by a calendar spread involving our ε-payoff function
gε(S, t).
StockAug2005 – p.16/40
Modifications needed for Lognormal Dynamics
{(S1t, S2t) : t ∈ [0, T ]}:
dS1t = S1tσ1atdW1t
dS2t = S2tσ2atdW2t, t ∈ [0, T ],
and initial spot prices S10 > 0 and S20 > 0. Here σ1 and σ2 are constants (this assumptionwill be relaxed later) and {W1t : t ∈ [0, T ]} and {W2t : t ∈ [0, T ]} are correlatedone-dimensional standard Brownian motions, with constant correlation coefficient ρ i.e.:
< W1t, W2t >t= ρt.
Letting xit = ln(Sit), i = 1, 2, and applying Itô’s formula, we may write the risk-neutraldynamics in the x variables as:
dxit = −σ2i
2a2
t dt + σiatdWit, t ∈ [0, T ], i = 1, 2.
StockAug2005 – p.17/40
Solution: Log-normal dynamics
Consider a function U(x1, x2) which solves the following canonical version of the two dimensionalconstant coefficient elliptic equation:
σ21
2
∂2
∂x21
U(x1, x2) + ρσ1σ2∂2
∂x1∂x2U(x1, x2) +
σ22
2
∂2
∂x22
U(x1, x2) + ν1∂
∂x1U(x1, x2) + ν2
∂
∂x2U(x
= −V atmI δ(x1 − k1, x2 − k2),
where ki ≡ ln Ki, i = 1, 2. A fundamental solution can be found in explicit form:
g(x, y, k) ≡ VI
π√4
exp
[
−1
2νtA(x − k)
]
K0
[
Q√
(x − k)tA(x − k)]
,
where 4 again denotes the determinant of the covariance matrix and ν is the vector of drifts:
ν = [2ν1, 2ν2] =[−σ2
1 ,−σ22
]
the matrix A ≡ a−1 is the inverse of a, K0 is the modified Bessel function of the second kind and
degree zero, and Q is the constant νtAν.StockAug2005 – p.18/40
Higher dimension: Lognormal dynamics
Constant coefficients: Everything generalizes naturally.
StockAug2005 – p.19/40
Variable coefficients
Let us consider again the stochastic differential equations:
dSit = atσijdWijt, i = 1, . . . , n,
where now σij = σij(S1, . . . , Sn) can depend on the stock prices S, but not on time. Withaij = Trace(σσt), and x = ln(S).
We again are lead to consider the following variable coefficient elliptic equation:
Lu = aij(x1, . . . , xn)uxixj + νi(x1, . . . , xn)uxi = −δ(x − k), (10)
in the log variables xi = ln Si, ki = ln Ki, i = 1, . . . , n. Here, as before, νi = −n∑
i=1
σ2ij
2.
In order to explain Hadamard’s approach, we need to recall the idea of geodesic distance.
StockAug2005 – p.20/40
Riemannian
0.1 Riemannian Metrics and Geodesic Distance
Consider the inverse A = a−1 of the coefficient matrix a. A Riemannian metric is defined by:
ds2 = Aijdxidxj . (11)
The geodesics are curves x(τ) = (x1(τ), . . . , xn(τ)) that minimize the integral:
I =
∫ τ1
τ0
√√√√
n∑
i,j=1
Aij xixjdτ
between fixed limits:
x(τ0) = x, x(τ1) = x.
Consider the square Γ = I2 of the geodesic distance as a function of its upper limit x. It can beshown that this function solves the following first order partial differential equation:
aijΓxiΓxj = 4Γ. (12)StockAug2005 – p.21/40
Riemannian ct’d
Setting pi = Γxi , i, . . . , n, the Hamiltonian associated to this PDE is:
H =1
4aijpipj ,
geodesics are determined by integrating the system of so-called ray equations:
dxi
dt=
1
2aijpj ,
dpi
ds= −1
4
∂ajk
∂xi
pipk, i = 1, . . . , n,
subject to the initial conditions:
xi(0) = xi, pi(0) = qi.
The qi parametrize the direction of the tangent line to the geodesic through the base point x
in the tangent plane to the surface defined by the Riemannian surface associated to ourmetric Aijdsidsj .
StockAug2005 – p.22/40
Hadamard’s Method for Determining the Fundamental
We wish to solve
aij(x, t)uxixj + bi(x, t)uxi = −δ(x − K),
i.e. to determine a fundamental solution of the elliptic equation with variable coefficients.
There are several methods.
Perhaps the most geometrically appealing and most constructive in flavour is Hadamard’smethod.(Brief Introduction to Hadamard’s Method)
Recall the Riemannian metric associated to the principal part of the el liptic differential operator . Ie.
Aijdsidsj
where
a = a−1 a = {aij}StockAug2005 – p.23/40
Hadamard’s Method Ct’d
There are some technical differences in the method according as to whether the underlyingspatial dimension is even or odd. For reasons of brevity, here we limit our description to thecase of an odd dimension.
( Odd n ≥ 3 )
F =U
Γn−2
2
, (13)
where m ≡ n−22
and where U is an infinite series of the following form:
U0 + U1Γ + U2Γ2 + . . . =
+∞∑
l=1
UlΓl.
where Γ is the square of the geodesic distance in the Riemannian metric defined above.
StockAug2005 – p.24/40
System of ODE’s along geodesics
Plugging this ansatz into the PDE and collecting terms multiplying the same power of Γ
leads to the iterative determination of solution by solving system of ODE’sWe have:
sdU0
ds+
M − 2n
4U0 = 0
4sdUj
ds+ (M + 4j − 2n)Uj − LUj−1
j − m= 0, j ≥ 1.
By means of the substitution y = ln s and using an integrating factor, the second set ofequations can be written as:
d
ds
(sjdUj
U0
)
=sj−1LUj−1
4(j−m)U0, l ≥ 1,
where here: .
M ≡ −L(Γ).
StockAug2005 – p.25/40
Solution
The solution of these equations can be iteratively determined in the form:
U0(x, x′) =γ(m)
4πn2
exp
(
−∫ s
0
M(x(u), y) − 2n
4udu
)
Uj(x, y) =U0(x, y)
4(j − m)sj
∫ s
0
uj−1(LUj−1)(x(u), y)
U0(x, y)du, j ≥ 1.
Here, x(u) denotes the path on the unique geodesic joining y and x.
Hadamard shows that for small enough neighborhood of center the resulting seriesconverges, provided the coefficients of the solution are assumed to be analytic.
The geodesic neighborhood must also be taken to be sufficiently small so as to ensure thatgeodesics do not cross.
StockAug2005 – p.26/40
Idea of Construction of gε payoff in the simplest
.We will illustrate the construction in the simplest 2 − D case where the stocks follow anormal dynamics.
Idea of construction is a natural extension of a similar construction used in one of the proofsof the Meyer-Tanaka formula for local time.
(Tanaka ) Generalized Ito’s formula for convex functions with corner. For illustration considersimpler Tanaka’s formula for taking stochastic differential of g(Wt) where
f(x) = |x|
It says (we saw it at the beginning of the talk):
|Wt| = |W0| +∫ t
0sign(Ws)dWs + Lt(ω)
where Lt, the local time at zero defined by
Lt = limε→0
1
2ε|{s ∈ [0, t]; Ws ∈ (−ε, ε)] limit in L2(P ) sense
StockAug2005 – p.27/40
Approximating function
One possible proof of Tanaka formula approximates |x| by aquadratic function in the region |x| ≤ ε.
gε(x) =
{
|x| if |x| ≥ ε1
2(ε + x
2
2) if |x| < ε
StockAug2005 – p.28/40
Illustration Tanaka formula
parabola in this region
f(x) = |x|
StockAug2005 – p.29/40
Approximating Tanaka formula
gε(x) approximating function we obtain the approxating formula
gε(Wt) = gε(W0) +
∫ t
0(gε)′(Ws)dWs +
1
2ε|{s ∈ [0, t]; Ws ∈ (−ε, ε)}|
What is the analogue to this approach in dimension n ≥ 2 ?Answer: Replace the fundamental solution inside the ellipsoid by a multivariate quadraticfunction in such a way that:i) The function and its first derivatives paste smoothly along the boundary of the ellipsoid
ii) The first derivatives are piecewise C1 and the jumps in the second derivatives occur butthese are globally bounded.
For n = 2 such a quadratic function can be shown to be
g(x, ξ) = − 1
4π√
∆ε2A2(x, ξ) +
1
4π√
∆(1 − ln(ε2)) for n = 2
where A2(x, ξ) = −2∑
i,j=1Aij(xi − ξi) (recall A = a−1).
StockAug2005 – p.30/40
Radon-Transform Approach to multi-D Breeden
Payoff of a basket option:
(∑
wiSi − K)+
Normalize this payoff
|w|(∑
w′iSi − q)+ where ξi =
wi
|w| , q =K
|w|
Consider C(ξ, q) i.e. the prices of all basket options with ξ ∈ Sn−1, q ∈ [0,∞] . Letp(S1, · · · , Sn) denote the risk-neutral density, so that
C(ξ, q) =
∫
<+
(ξ · S − q)+p(S1, · · · , Sn)dS1 · · · dSn
StockAug2005 – p.31/40
Radon Ct’d
∂2
∂q2C(ξ, q) =
∫
S·ξ−q=0
p(S1, · · · , Sn)dS1 · · · dSn
ie. C(ξ, q) corresponds to the integral of density p on a hyperplane with normal ξ ∈ Sn−1
and a distance q from the origin.
∂2
∂q2C(ξ, q) = R[p](ξ, q)
where R[p](ξ, q) is the Radon transform of the density p.
StockAug2005 – p.32/40
Radon-Conclusion
Now key observation is that there is an inversion formula for the Radon transform whichreads
v(ξ, q) = R[p](ξ, q)
⇔
p(S1, · · ·Sn) =
∫
|ξ|=1h(S · ξ, ξ)dξ
Here definition of h depends on whether dimension is n is even or odd:
h(q, ξ) ≡ (i)n−1
2(2π)n−1
∂n−1
∂qn−1R[f ](ξ, q) n odd
h(q, ξ) ≡ (i)n
2(2π)n−1H
[∂n−1
∂pn−1R[f ](ξ, q)
]
(q) n even
where H[g(p)](t) denotes the Hilbert transform of the function g(p):
H[g(p)](t) ≡ 1
π
∫ ∞
−∞
g(p)
p − tdp, (14)
StockAug2005 – p.33/40
Closed form solutions for option pricingand interest models on two assets
– p.1/58
The result is to appear in
International Journal of Theoretical and Applied Finance.
– p.2/58
Black-Scholes equation
∂C
∂τ+σ2
2S2
∂C2
∂S2+ rS
∂C
∂S− rC = 0
t = T − τ
∂C
∂t=σ2
2S2
∂C2
∂S2+ rS
∂C
∂S− rC
ξ = lnS
∂C
∂t=σ2
2
∂C2
∂ξ2+
(
r − σ2
2
)
∂C
∂ξ− rC
– p.3/58
Black-Scholes equation
∂C
∂τ+σ2
2S2
∂C2
∂S2+ rS
∂C
∂S− rC = 0
t = T − τ
∂C
∂t=σ2
2S2
∂C2
∂S2+ rS
∂C
∂S− rC
ξ = lnS
∂C
∂t=σ2
2
∂C2
∂ξ2+
(
r − σ2
2
)
∂C
∂ξ− rC
– p.3/58
Black-Scholes equation
∂C
∂τ+σ2
2S2
∂C2
∂S2+ rS
∂C
∂S− rC = 0
t = T − τ
∂C
∂t=σ2
2S2
∂C2
∂S2+ rS
∂C
∂S− rC
ξ = lnS
∂C
∂t=σ2
2
∂C2
∂ξ2+
(
r − σ2
2
)
∂C
∂ξ− rC
– p.3/58
Black-Scholes equation
c(ξ, t) = ertC(ξ, t)
∂c
∂t=σ2
2
∂c2
∂ξ2+
(
r − σ2
2
)
∂c
∂ξ
x = ξ +(
r − σ2
2
)
t
∂c
∂t=σ2
2
∂2c
∂x2
– p.4/58
Black-Scholes equation
c(ξ, t) = ertC(ξ, t)
∂c
∂t=σ2
2
∂c2
∂ξ2+
(
r − σ2
2
)
∂c
∂ξ
x = ξ +(
r − σ2
2
)
t
∂c
∂t=σ2
2
∂2c
∂x2
– p.4/58
Black-Scholes equation
In total, we have done the transformation
t = T − τ
x = lnS +
(
r − σ2
2
)
(T − τ)
c = er(T−τ)C
which transforms Black-Scholes equation to heat equation.
– p.5/58
Merton
Merton’s arguments in his 1973 paper imply moregenerally that the arbitrage-free value C of manyderivatives satisfies
∂C
∂τ+σ2(S, τ)
2S2
∂C2
∂S2+ b(S, τ)
∂C
∂S− r(S, τ)C = 0
with three variable coefficients σ(S, τ), b(S, τ) and r(S, τ).
Carr-Lipton-Madan(2000) characterized the entire set ofvariable coefficients {σ(S, τ), b(S, τ), r(S, τ)} which permitthe above one state variable valuation PDE to betransformed to the heat equation.
– p.6/58
Merton
Merton’s arguments in his 1973 paper imply moregenerally that the arbitrage-free value C of manyderivatives satisfies
∂C
∂τ+σ2(S, τ)
2S2
∂C2
∂S2+ b(S, τ)
∂C
∂S− r(S, τ)C = 0
with three variable coefficients σ(S, τ), b(S, τ) and r(S, τ).
Carr-Lipton-Madan(2000) characterized the entire set ofvariable coefficients {σ(S, τ), b(S, τ), r(S, τ)} which permitthe above one state variable valuation PDE to betransformed to the heat equation.
– p.6/58
Group classification
Lie’s group classification of linear second order PDE with 2independent variables
Put +Qux +Ruxx + Su = 0
P,R 6= 0 and P,Q,R, S are functions of t and x.
Any equation is admitted by the generators of the trivial
symmetries u∂
∂uand φ(t, x)
∂
∂u. Any equation can be
reduced to the form
vτ = vyy + Z(τ, y)v
by a transformation as y = α(t, x), τ = β(t) and v = γ(t, x)u
with αx 6= 0 and βt 6= 0.– p.7/58
Group classification
An equation admits symmetry generators generated by
u∂
∂u, φ(t, x)
∂
∂u,
∂
∂τ
can be reduced to the form
vτ = vyy + Z(y)v
by a transformation.
– p.8/58
Group classification
An equation admits symmetry generators generated by
u∂
∂u, φ(t, x)
∂
∂u,
∂
∂τ, 2τ
∂
∂τ+ y
∂
∂y
τ2∂
∂τ+ τy
∂
∂y−(
1
4y2 +
1
2τ
)
v∂
∂v
can be reduced to the form
vτ = vyy +A
y2v
by a transformation, where A is a constant.
– p.9/58
Group classification
An equation admits symmetry generators generated by
u∂
∂u, φ(t, x)
∂
∂u,
∂
∂τ2τ
∂
∂τ+ y
∂
∂y,
τ2∂
∂τ+ τy
∂
∂y−(
1
4y2 +
1
2τ
)
v∂
∂v
∂
∂y, 2τ
∂
∂y− yv
∂
∂v
can be reduced to the form
vτ = vyy
by a transformation.
– p.10/58
Convenient Variables
The main difficulty in integrating a given differential equationlies in introducing convenient variables, which there is norule for finding. Therefore we must travel the reverse pathand after finding some notable substitution, look forproblems to which it can be successfully applied.
- Jacobi, Lectures on Dynamics, 1847.
– p.11/58
Lie’s Symmetry Analysis
From geometric point of view, the great Norwegian mathe-
matician Sophus Lie created a systematic way for analyzing
solutions of differential equations. His theory nowadays has
been known as Lie’s symmetry analysis of differential equa-
tion. Up to now, Lie’s theory has been greatly extended and
used to find closed form solutions, to classify equations up to
some unknown functions, to find fundamental solutions and
so on.
– p.12/58
Symmetry group
A symmetry group of a system S of differential equations isa local group of transformations, G, acting on some opensubset M ⊂ X × U in such a way that "G transformssolutions of S to other solutions of S".
Allowing arbitrary nonlinear transformations of both theindependent and dependent variables in the definition ofsymmetry.
– p.13/58
Prolongation of f
f : X → U , u = f(x) u(n) := pr(n)f(x) : X → U (n): the n-th
prolongation of f is defined by
uJ = ∂Jf(x)
For example, u = f(x, y), the second prolongationu(2) = pr
(2)f(x, y) is given by
(u;ux, uy;uxx, uxy, uyy) =
(
f ;∂f
∂x,∂f
∂y;∂2f
∂x2,∂2f
∂x∂y,∂2f
∂y2
)
.
– p.14/58
Solutions of DE’s
An n-th order differential equation in p independentvariable is given as
∆(x, u(n)) = 0
involving x = (x1, · · · , xp), u and the derivatives of u withrespect to x up to order n.
∆ can be viewed as a function from the jet spaceX × U (n) to R, i.e., ∆ : X × U (n) → R.
Hence, an n-th order differential equation can beregarded as a subvariety S∆ := {∆(x, u(n)) = 0} in then-jet space X × U (n).
– p.15/58
Solutions of DE’s
A smooth solution of the given n-th order differentialequation in this sense is a smooth function u = f(x) suchthat
∆(x,pr(n)f(x)) = 0
whenever x lies in the domain of f .
Hence a solution can be regarded as a function u = f(x)
such that the graph of the n-th prolongation pr(n)f is
contained in the subvariety S∆, i.e.,
Γ(n)f := {(x,pr
(n)f(x))} ⊂ S∆
– p.16/58
Prolongation of group actions
G: local group of transformations acting on M ⊂ X × U
The n-th prolongation pr(n)G of G is an induced local
action of G on the n-jet space M (n) which is defined sothat it transforms the derivatives of functions u = f(x)
into the corresponding derivatives of the transformedfunction u = f(x).
This can be done by calculating the transformation ofn-th order Taylor polynomial.
– p.17/58
Invariance of DEs
∆(x, u(n)) = 0: n-th order differential equation definedover M
S∆ ⊂M (n): corresponding subvariety
G: local group of transformations acting on M whoseprolongation leaves S∆ invariant, i.e.,
(x, u(n)) ∈ S∆ ⇒ pr(n)g · (x, u(n)) ∈ S∆ for all g ∈ G.
G is a symmetry group of the differential equation !
– p.18/58
Invariance of DEs
∆(x, u(n)) = 0: n-th order differential equation definedover M
S∆ ⊂M (n): corresponding subvariety
G: local group of transformations acting on M whoseprolongation leaves S∆ invariant, i.e.,
(x, u(n)) ∈ S∆ ⇒ pr(n)g · (x, u(n)) ∈ S∆ for all g ∈ G.
G is a symmetry group of the differential equation !
– p.18/58
Prolongation of vector fields
v: vector field on M
exp(εv): corresponding one-parameter group
The n-th prolongation pr(n)
v of v is a vector field on the jetspace M (n) and defined to be the infinitesimal generator ofthe corresponding prolonged group pr
(n)[exp(εv)], i.e.,
pr(n)
v|(x,u(n)) =d
dε
∣
∣
∣
∣
ε=0
pr(n)[exp(εv)](x, u(n))
for any (x, u(n)) ∈M (n).
– p.19/58
Infinitesimal invariance
Lie’s Theorem∆(x, u(n)): n-th order differential equation
Let v be a vector field. Then v generates a oneparameter local group of symmetries if and only if
pr(n)
v[∆(x, u(n))] = 0
whenever ∆(x, u(n)) = 0.
The set of all infinitesimal symmetries form a Lie algebraof vector fields on X × U . Moreover, if this Lie algebra isfinite dimensional, the symmetry group of the equation isa local Lie group of transformations acting on X × U .
– p.20/58
Prolongation formula
The second prolongation pr(2)
v of a vector field v with
v =∑
i
ξi(t, x, u)∂
∂xi+ φ(t, x, u)
∂
∂u
is as
pr(2)
v = v +∑
i
φi ∂
∂uxi+∑
i,j
φij ∂
∂uxixj
where the φ’s are given by the prolongation formulas
– p.21/58
Total differentiation
φi = Di(φ)−∑
j
uxjDi(ξj)
φij = Dj(φi)−
∑
k
uxixkDj(ξk)
and the D’s are the so-called total differentiation.
Di =∂
∂xi+ uxi
∂∂u + uxixk
∂∂u
xk+ · · ·
– p.22/58
Determining equation
The determining equation
pr(n)
v[∆(x, u(n))] = 0 whenever ∆(x, u(n)) = 0
is a linear homogeneous partial differential equation of thesecond order for the unknown functions ξ, τ, φ of the"independent variables" t, x, u.
t, x, u, uxi , uxixj , · · · , are regarded as "independent" ones.
The determining equation decomposes into a system ofseveral equations.
The resulting system is an overdetermined system whichcan be solved analytically.
– p.23/58
Determining equation
The determining equation
pr(n)
v[∆(x, u(n))] = 0 whenever ∆(x, u(n)) = 0
is a linear homogeneous partial differential equation of thesecond order for the unknown functions ξ, τ, φ of the"independent variables" t, x, u.
t, x, u, uxi , uxixj , · · · , are regarded as "independent" ones.
The determining equation decomposes into a system ofseveral equations.
The resulting system is an overdetermined system whichcan be solved analytically.
– p.23/58
Determining equation
The determining equation
pr(n)
v[∆(x, u(n))] = 0 whenever ∆(x, u(n)) = 0
is a linear homogeneous partial differential equation of thesecond order for the unknown functions ξ, τ, φ of the"independent variables" t, x, u.
t, x, u, uxi , uxixj , · · · , are regarded as "independent" ones.
The determining equation decomposes into a system ofseveral equations.
The resulting system is an overdetermined system whichcan be solved analytically.
– p.23/58
Determining equation
The determining equation
pr(n)
v[∆(x, u(n))] = 0 whenever ∆(x, u(n)) = 0
is a linear homogeneous partial differential equation of thesecond order for the unknown functions ξ, τ, φ of the"independent variables" t, x, u.
t, x, u, uxi , uxixj , · · · , are regarded as "independent" ones.
The determining equation decomposes into a system ofseveral equations.
The resulting system is an overdetermined system whichcan be solved analytically.
– p.23/58
1 dim heat equation
ut = uxx
The heat equation can be identified with the linearsubvariety in X × U (2) determined by the vanishing of∆(x, t, u(2)) = ut − uxx.
v = ξ(t, x, u) ∂∂x + τ(t, x, u) ∂∂t + φ(t, x, u) ∂
∂u
pr(2)
v = v + φx ∂∂ux+ φt ∂
∂ut+ φxx ∂
∂uxx+ φxt ∂
∂uxt+ φtt ∂
∂utt
– p.24/58
Infinitesimal invariance
pr(2)v[∆(x, t, u(2))] = φt − φxx = 0
φt = φt − ξtux + (φu − τt)ut − ξuuxut − τuu2t
φxx = φxx + (2φxu − ξxx)ux − τxxut + ξuuu3x − τuuu
2xut + (φu −
2ξx)uxx − 2τxuxt − 3ξuuxuxx − τuutuxx − 2τuuxuxtReplacing ut by uxx
– p.25/58
Determining equation
Monomial Coefficient
uxuxt −2τu = 0uxt −2τx = 0u2xx −τu = −τuu2xuxx 0 = −τuuuxuxx −ξu = −2τxu − 3ξuuxx φu − τt = −τxx + φu − 2ξxu3x 0 = −ξuuu2x 0 = φuu − 2ξxuux −ξt = 2φxu − ξxx
1 φt = φxx
– p.26/58
General solution
ξ(x, t) = c1 + c4x+ 2c5t+ 4c6xt
τ(t) = c2 + 2c4t+ 4c6t2
φ(x, t, u) = (c3 − c5x− 2c6t− c6x2)u+ α(x, t)
c1, · · · , c6 are arbitrary constants and α(x, t) an arbitrarysolution of the heat equation.
– p.27/58
Infinitesimal symmetries
Finite (six) dimensional subalgebra
v1 = ∂x
v2 = ∂t
v3 = u∂u
v4 = x∂x + 2t∂t
v5 = 2t∂x − xu∂u
v6 = 4tx∂x + 4t2∂t − (x2 + 2t)u∂u
Infinite dimensional subalgebra
vα = α(x, t)∂u,
where α is an arbitrary solution of the heat equation.– p.28/58
Symmetry group
G1 : (x+ ε, t, u)
G2 : (x, t+ ε, u)
G3 : (x, t, eεu)
G4 : (eεx, e2εt, u)
G5 : (x+ 2εt, t, u · exp(−εx− ε2t))
G6 :
(
x
1− 4εt ,t
1− 4εt , u√1− 4εte −εx
2
1−4εt
)
Gα : (x, t, u+ εα(x, t))
– p.29/58
Symmetry group
G3, Gα: the linearity of the heat equation
G1, G2: time and space invariance of the equation,reflecting that heat equation is of constant coefficients.
G4: well-known scaling symmetry
G5: Galilean boost to a moving coordinate frame
G6: fundamental solution
– p.30/58
Solutions from symmetry
If u = f(x, t) is a solution of the heat equation, so are thefollowing functions
u1 = f(x− ε, t)
u2 = f(x, t− ε)
u3 = eεf(x, t)
u4 = f(e−εx, e−2εt)
u5 = e−εx+ε2tf(x− 2εt, t)
u6 =1√1 + 4εt
e−εx21+4εtf(
x
1 + 4εt,
t
1 + 4εt)
uα = f(x, t) + εα(x, t)
– p.31/58
Finkel
Finkel has classified the 2 dimensional parabolicequations of the form
ut −1
2∆u+M(x, y)u = 0.
Symmetry operators in the following are of the form
X = τ(t)∂t + ξ(x, y, t)∂x + η(x, y, t)∂y + ϕ(x, y, t)u∂u
where
ξ =τt
2+ γy + ξ0(t)
η =τt
2− γx+ η0(t)
ϕ = − τtt4(x2 + y2)− ξ′0x− η′0y + φ(t)
– p.32/58
Classification by Finkel - 1.1a
Case 1.1a, Dimension of group = 4
M =C0
x2+ by + c0, C0 6= 0
τ = δ2t2 + δ1t, γ = ξ0 = 0, η0 =
bδ2
2t3 +
3bδ1
4t2 + β1t+ β0,
φ = − b2δ2
8t4 − b2δ1
4t3 − ( bβ1
2+ c0δ2)t
2 − (δ2 + c0δ1 + bβ0)t
– p.33/58
Classification by Finkel- 1.1b
Case 1.1b, Dimension of group = 4
M =C0
x2+ cr2 + by + c0, C0 6= 0
τ = δ1e2√
2ct + δ2e−2√
2ct, γ = ξ0 = 0,
η0 =bδ1√2ce2√
2ct − bδ2√2ce−2
√2ct + β1e
√2ct + β2e
−√
2ct,
φ = −(√
2c+ c0 +b2
4c
)
δ1e2√
2ct +
(√2c− c0 −
b2
4c
)
δ2e−2√
2ct
− bβ1√2ce√
2ct +bβ2√2ce−√
2ct
– p.34/58
Classification by Finkel- 1.2a
Case 1.2a, Dimension of group = 2
M = C(θ)1
r2+ c0
τ = δ2t2 + δ1t, γ = ξ0 = η0 = 0, φ = −c0δ2t2 − (δ2 + c0δ1)t,
where C(θ) 6= (C0 cos θ + C1 sin θ)−2 and C ′ 6= 0.
– p.35/58
Classification by Finkel- 1.2b
Case 1.2b, Dimension of group = 2
M = C(θ)1
r2+ cr2 + c0
τ = δ1e2√2ct + δ2e
−2√2ct, γ = ξ0 = η0 = 0,
φ = −(√2c+ c0
)
δ1e2√2ct +
(√2c− c0
)
δ12e−2√2ct
with C(θ) 6= (C0 cos θ + C1 sin θ)−2 and C ′ 6= 0.
– p.36/58
Classification by Finkel- 1.3
Case 1.3, Dimension of group = 1
M = C(λ log r + θ)1
r2+ c0, C ′ 6= 0 6= λ
τ =2γ
λt, ξ0 = η0 = 0, φ = −2c0γ
λt
– p.37/58
Classification by Finkel- 1.4a
Case 1.4a, Dimension of group = 3
M = C01
r2+ ax+ by + c0, C0 6= 0
τ = δ2t2 + δ1t, ξ0 = η0 = 0, φ = −c0δ2t2 − (δ2 + c0δ1)t
with the constraints δ1 = δ2 = 0 if a 6= 0 or b 6= 0.
– p.38/58
Classification by Finkel- 1.4b
Case 1.4b, Dimension of group = 3
M = C0r−2 + cr2 + ax+ by + c0, C0 6= 0
τ = δ1e2√2ct + δ2e
−2√2ctb, ξ0 = η0 = 0,
φ = −(√2c+ c0)δ1e
2√2ct + (
√2c− c0)δ2e
−2√2ct
with the constraint δ1 = δ2 = 0 if a 6= 0 or b 6= 0.
– p.39/58
Classification by Finkel- 1.5a
Case 1.5a, Dimension of group = 7
M = ax+ by + c0
τ = δ2t2 + δ1t,
ξ0 =aδ2
2t3 +
1
4(3aδ1 − 2bγ)t2 + α1t+ α0,
η0 =bδ2
2t3 +
1
4(3bδ1 + 2aγ)t
2 + β1t+ β0,
φ = −18(a2 + b2)δ2t
4 − 1
4(a2 + b2)δ1t
3 − ( 12(aα1 + bβ1) + c0δ2)t
2
−(δ2 + c0δ1 + aα0 + bβ0)t
– p.40/58
Classification by Finkel- 1.5b
Case 1.5b, Dimension of group = 7
M = cr2 + ax+ by + c0
τ = δ1e2√
2ct + δ2e−2√
2ct,
ξ0 =aδ1√2ce2√
2ct +aδ2√2ce−2
√2ct + α1e
√2ct + α2e
−√
2ct +bγ
2c,
η0 =bδ1√2ce2√
2ct +bδ2√2ce−2
√2ct + β1e
√2ct + β2e
−√
2ct − aγ
2c,
φ = −(√
2c+ c0 +a2 + b2
4c
)
δ1e2√
2ct +
(√2c− c0 −
a2 + b2
4c
)
δ2e−2√
2ct
−aα1 + bβ1√2c
e√
2ct +aα2 + bβ2√
2ce−√
2ct
– p.41/58
Classification by Finkel- 1.6
Case 1.6, Dimension of group = 1
M = C(r) + dθ
τ = ξ0 = η0 = 0, φ = dγt
with C(r) 6= C0r−2 + C1r
2 + c0 if d = 0.
– p.42/58
Classification by Finkel- 1.7a
Case 1.7a, Dimension of group = 4
M = cx2 + ax+ by + c0
τ = γ = 0, ξ0 = α1e√2ct + α2e
−√2ct, η0 = β1t+ β0,
φ = − aα1√2ce√2ct +
aα2√2ce−√2ct − bβ1
2t2 − bβ0t
– p.43/58
Classification by Finkel- 1.7b
Case 1.7b, Dimension of group = 4
M = c(x2 − y2) + ax+ by + c0
τ = γ = 0, ξ0 = α1e√
2ct + α2e−√
2ct, η0 = β1e√−2ct + β2e
−√−2ct,
φ = − aα1√2ce√
2ct +aα2√2ce−√
2ct − bβ1√−2c
e√−2ct +
bβ2√−2c
e−√−2ct
– p.44/58
Classification by Finkel- 1.8a
Case 1.8a, Dimension of group = 2
M = C(x) + by
τ = γ = ξ0 = 0, η0 = β1t+ β0, φ = −bβ12t2 − bβ0
with C0x2 + ax+ c0 6= C(x) 6= C0
x2 + c0.
– p.45/58
Classification by Finkel- 1.8b
Case 1.8b, Dimension of group = 2
M = C(x) + cy2 + by
τ = γ = ξ0 = 0, η0 = β1e√2ct + β2e
−√2ct,
φ = − bβ1√2ce√2ct +
bβ2√2ce−√2ct
with C0x2 + ax+ c0 6= C(x) 6= C0
x2 + cx2 + c0.
– p.46/58
Finding Fundamental Solutions
The fundamental solutions are determined by
Restricting the symmetry operator of the equationut − 1
2∆u+Mu = 0 to the initial value problem{
ut − 12∆u+Mu = 0
u(x, 0) = δξ(x)
Finding the invariant solutions to the restricted symmetryoperators.
– p.47/58
Convention
The following convention are used in the following slides
Sign convention
± ={
+ when c > 0
− when c < 0
Notation convention
cot± =
{
coth(√2ct) when c > 0
cot(√−2ct) when c < 0
,
and similarly for sin±, sec±, csc± and tan±.
– p.48/58
Fundamental Solution - 1.1a
ut − 12∆u+ (
C0
x2 + by + c0)u = 0
Fundamental Solution (Transition probability)
G(1a) = K( x
t
)κ e−2 ξxt−c0t
tΦ
(
κ, 2κ, 2ξx
t
)
×
exp
{
− (x− ξ)2 + (y − η)2
2t− b
2(y + η)t+
b2
24t3}
,
κ = 1+√1+8C0
2
Φ(κ, 2κ, z) is a confluent hypergeometric function of order (κ, 2κ), i.e., Φ satisfies thedifferential equation zv′′ + (2κ− z)v′ − κv = 0, with the asymptotic behavior
Φ→ Γ(2κ)
Γ(κ)ezz−κ as z → +∞.
– p.49/58
Fundamental Solution - 1.1b
ut − 12∆u+ (
C0
x2 + c(x2 + y2) + by + c0)u = 0
Fundamental Solution (Transition probability)
G(1b) = K(x csc±)κeγx csc± −c0t csc± Φ(κ, 2κ,−2γx csc±)×
exp
{
−√
|c|2cot±(x2 + ξ2)
}
×
exp
[
− cot±{
√
|c|2(y − η sec±)2 ± b
√
2|c|(1− sec±)(y − η sec±)
}]
×
exp
[
b2
4|c|
(√
2
|c| (csc±− cot±)± t
)
−(
bη√
2|c|±√
|c|2(η2)
)
tan±]
– p.50/58
Fundamental Solution - 1.2a
ut − 12∆u+ (C(θ)
1r2 + c0)u = 0
Fundamental Solution (Transition probability) at (0, 0)
G(2a) =Ke−c0t
te−
x2+y2
2t l(θ)
where l is a solution to
l′′(θ) = 2C(θ)l(θ) for θ ∈ [0, 2π]
with the boundary condition l(0) = l(2π)
– p.51/58
Fundamental Solution - 1.2b
ut − 12∆u+ (C(θ)
1r2 + c(x2 + y2) + c0)u = 0
Fundamental Solution (Transition probability) at (0, 0)
G(2b) =Ke−c0t
sin±exp
[
−√
|c|2cot±(x2 + y2)
]
l(θ),
where l is a solution to
l′′(θ) = 2C(θ)l(θ) for θ ∈ [0, 2π]
with the boundary condition l(0) = l(2π)
– p.52/58
Fundamental Solution - 1.4a
ut − 12∆u+ (
C0
r2 + c0)u = 0
Fundamental Solution (Transition probability) at (0, 0)
G(4a) =Ke−c0t
t
(
√
x2 + y2
t
)λ
Φ
(
1 + λ, 1 + λ,−x2 + y2
2t
)
λ =√2C0
Φ is the incomplete Gamma functionΓ(−λ, z) =
∫∞z
t−λ−1e−tdt.
– p.53/58
Fundamental Solution - 1.4b
ut − 12∆u+ (
C0
r2 + c(x2 + y2) + c0)u = 0
Fundamental Solution (Transition probability) at (0, 0)
G(4b) =Ke−c0t
sin±
(
√
x2 + y2
sin±
)λ
e−
√
|c|2(x2+y2) cot±
,
where λ =√2C0
– p.54/58
Fundamental Solution - 1.5a
ut − 12∆u+ (ax+ by + c0)u = 0
Fundamental Solution (Transition probability)
G(5a) =Ke−c0t
texp
[
− (x− ξ)2 + (y − η)2
2t−(
a
2(x+ ξ) +
b
2(y + η)
)
t+a2 + b2
24t3]
– p.55/58
Fundamental Solution - 1.5b
ut − 12∆u+ (c(x
2 + y2) + ax+ by + c0)u = 0
Fundamental Solution (Transition probability)
G(5b) =Ke−c0t
sin±×
exp
[
a2 + b2
4|c|
(√
2
|c| (csc±− cot±)± t
)
−(
aξ + bη√
2|c|±√
|c|2(ξ2 + η2)
)
tan±]
×
exp
[
− cot±{
√
|c|2(x− ξ sec±)2 ± a
√
2|c|(1− sec±)(x− ξ sec±)
}]
×
exp
[
− cot±{
√
|c|2(y − η sec±)2 ± b
√
2|c|(1− sec±)(y − η sec±)
}]
– p.56/58
Fundamental Solution - 1.7a
ut − 12∆u+ (cx
2 + ax+ by + c0)u = 0
Fundamental Solution (Transition probability)
G(7a) =Ke−c0t√t√sin±
exp
[
a2
4|c|
(√
2
|c| (csc±− cot±)± t
)
−(
aξ√
2|c|±√
|c|2ξ2
)
tan±
+b2t3
24− b
2(y + η)t
]
×
exp
[
− cot±{
√
|c|2
(
x− ξ sec±)2 ± a
√
2|c|(
1− sec±)
(x− ξ sec±)
}
− (y − η)2
2t
]
.
– p.57/58
Fundamental Solution - 1.7b
ut − 12∆u+ (c(x
2 − y2) + ax+ by + c0)u = 0
Fundamental Solution (Transition probability)
G(7b) =Ke−c0t√sin∓
√sin±
×
exp
[
a2
4|c|
(√
2
|c| (csc±− cot±)± t
)
−(
aξ√
2|c|±√
|c|2ξ2
)
tan±]
×
exp
[
b2
4|c|
(√
2
|c| (csc∓− cot∓)∓ t
)
−(
bη√
2|c|∓√
|c|2η2
)
tan∓]
×
exp
[
− cot±{
√
|c|2
(
x− ξ sec±)2 ± a
√
2|c|(
1− sec±)
(x− ξ sec±)
}]
×
exp
[
− cot∓{
√
|c|2
(
y − η sec∓)2 ∓ b
√
2|c|(
1− sec∓)
(y − η sec∓)
}]
– p.58/58
Topic I
aper I: Closed form solution for quadratic and inverse quadratic mod els of the term structure
ut − ∆u + Pu = 0, fundamental PDE or canonical form
where P is a potential.Typical examples of potentials arising in finance and in mathematicalphysics are
P = 0 heat equation
P = x21 + · + x2
n harmonic oscillator
P =λ
|x|2 inverse square potential
StockAug2005 – p.34/40
Topic I, ct’d
The reason that the PDE ut − ∆u + Pu = 0 is so importantderives from it’s being a prototype of a canonical form for asecond order parabolic differential equation with variable
coefficients . Indeed in the case of one spatial dimension it is the
canonical form. I.e.
Proposition 1 Any parabolic differential equation in one spatialvariable
ut −1
2a(x, t)uxx − b(x, t)ux + c(x, t)u = 0
can, by a change of independent and dependent variables, be re-
duced to canonical form.StockAug2005 – p.35/40
Topic 1, ct’d
This result goes back to Lie and it’s importance was redis-
covered in modern times by Bluman
StockAug2005 – p.36/40
Finkel’s classification of diffusions on<2 × [0, T ]
A Lie symmetry transformation is a transformation of theindependent variables x, y, t and dependent variable y of theform
(x, y, t, u) 7→ (x, y, t, u)
x = α1(x, y, t) y = α2(x, y, t)τ = β(t) v = γ(x, y, t)β(x, y, t
such that the equation
ut −1
2∆u + Pu = 0
is has the same form in the new variables.StockAug2005 – p.37/40
Use of Lie’s symmetry analysis
Lie’s symmetry analysis can be used to find similarityvariables which allow one, depending on the size of the Liesymmetry group to either
Reduce the PDE to an ODE.
Or
Integrate the PDE in closed form
StockAug2005 – p.38/40
Application of Finkel’s classification
An interesting application of Lie’s method is in expressing aparticular solution of the PDE in closed form, namely thefundamental solution.The application in mathematical finance is in expressing thefundamental solution of the zero coupon bond P (t, T )
P (t, T ) = E
[
exp
(∫ T
t
−r(X1, X2, t)ds
)]
in a situation where the pricing equation of P (t, T ) under theforward measure is for instance of the form
Pt −1
2∆P + a(t)X2
1 + b(t)X1X2 + c(t)X2
2 = 0,
a special case of the so-called quadratic term structuremodels. Much more general quadratic models have been
StockAug2005 – p.39/40
Conclusions
It is possible to define a multi-asset local variance contract.
In the context of multi-dimensional local volatility models, the construction of thecorresponding contingent claims involves solving for the fundamental solution of a variablecoefficient elliptic operator.
The construction is only local. Can it be extended ?
How many terms in the Hadamard expansion are needed to accurately approximate thefundamental solution in practice?
It is possible to extend Breeden and Litzenberger’s formula to n assets given the knowledgeof baskets with a continuum of strikes and of weights.
Since there are not sufficiently many weights and strikes, how can to best exploit thegeneralized Breeden and Litzenberger formula.
StockAug2005 – p.40/40